This invention relates to apparatus for calculating the amplitude value of a sinusoidal wave by sampling at a predetermined interval a sinusoidal voltage or current and then calculating the amplitude value of the voltage or current from a digital signal corresponding to the sampled value.
The calculating apparatus of this type are used for the remote control of the switches, measurement instruments, machines and apparatus of an electric power system. The digital signal corresponding to the sampled value is encoded and then transmitted to apparatus for calculating the amplitude value installed in a receiving station.
The following three methods of calculation have been used in the calculating apparatus.
The first method is an addition method wherein the absolute values of a plurality of data samples sampled in one half cycle of a sinusoidal voltage or current (hereinafter called an input alternating current) are added together and the result of addition is multiplied with a predetermined constant thereby determining the amplitude value of the input alternating current as shown in FIG. 1.
In FIG. 1 when an input alternating current having a frequency of 50 Hz is sampled at a sampling frequency of 600 Hz, the sampling period is 30.degree. so that the amplitude value can be determined by the following equation ##EQU1## WHERE M REPRESENTS A TIME SERIES, AND I.sub.K AN INSTANTANEOUS VALUE EXPRESSED BY THE FOLLOWING EQUATION (2). EQU i = I sin .omega. t. (2)
Taking m - 5 as the reference for the sampling points and assuming that the phase of the input alternating current is .omega. t, then the righthand side of equation (1) will be expressed as follows. ##EQU2##
In equation (3) the range of .omega. t is limited by an equation ##EQU3## so the righthand side of equation (3) will become as follows by taking into consideration the periodicity of the input alternating current. ##EQU4##
This equation shows that the error of the addition method caused by the variation in the sampling phase is less than .+-.1.7%.
The second method is a peak value detection method wherein data having the largest absolute value is selected among sampled data obtained in one half cycle or more and the selected data is used as the amplitude value. The selected maximum data shows the amplitude value of the input alternating current with an error in a definite range. Thus, for example, in the same manner as the addition method described above, when an input alternating current having a frequency of 50 Hz is sampled at a sampling frequency of 600 Hz the phase of the data having a maximum absolute value ranges from sin 5/12 .pi. to sin 1/2 .pi. so that the error caused by the variation of the sampling phase is less than .+-. 1.7% in the same manner as above described.
The third method is the square method. There is a formula regarding trigonometrical functions ##EQU5##
Thus, by availing the fact that the sum of squares of two sampling data dephased 90 electrical degrees is equal to the square of the amplitude value of the input alternating current, the amplitude value can be determined from the square root of said sum.
For example, similar to FIG. 1, when the input alternating current of 50 Hz is sampled at a sampling frequency of 600 Hz the square of the amplitude value can be determined by the following equation EQU I.sup.2 = i.sub.m.sup.2 + i.sub.m-3.sup.2 ( 7)
In principle, this square method is free from any error due to the variation in the sampling phase.
Thus, of these three methods the square method is most advantageous in that it does not accompany the error caused by the variation in the sampling phase, but this method is inconvenient since it is necessary to calculate a square root by using equation (7). When calculating a square root with an electronic computer it takes much longer time than an addition operation.
Although the addition method or the peak value detection method does not require such mathematical operation, it accompanies the error caused by the variation in the sampling phase.